Regulation Model of the Lake Velence Reservoir System
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REGULATION MODEL OF THE LAKE VELENCE RESERVOIR SYSTEM By K. KORIs-B. NAGY* Institute of Water Management and Hydraulic Engineering, Technical University, Budapest (Received: November 1, 1976) Presented by Prof Dr. 1. V. NAGY Introdnction A Hungarian water resources system conslstlllg of Lake Velence and two upstream reservoirs v,-ill be analyzed from the aspect of meeting optimum recreation conditions in the lake region. The integrate, complex development program of the Lake Velence recreation area requires the lake level to be regulated to meet recreation demands. Regulation will be achieved by the Zamoly reservoir, formerly complet ed, and by the recently constructed Patka reservoir. The two reservoirs and the {)utlet gate of Lake Velence "will maintain the optimum lake level over extended periods of time. A model has been established for estimating damages concomitant to lake levels other than ideal (high or low) to find the optimum schedule of operation, so as to minimize the average value of the damages resulting from level fluctuations, rather than the fluctuations themselves. In what fol lows the mathematical model and its practical application "\Y-ill be described. The water resources system has to be operated in compliance with {)bjectives set out before. During the past two or three decades, important advances have been made in the mathematical modelling of water resources systems [1, 2, 3,4, 5, 6]. These have proved successful, among others, in optimiz ing operating schedules. Description of the system Lake Velence, 'with an area of 26 sq· km, is situated in the western part of Hungary, not very distant from Lake Balaton. In its catchmen1 area of 615 sq.km two reservoirs have been constructed, namely the reservoirs Zamoly and Patka, v,-ith volumes of 4,5 ·10' and 8.102 cu· m, respectively (Fig. 1). * Department of C. E. Mathematics, Technical University, Budapest 148 KORIS-NAGY lE () 1 2 3 it 5 km .. -*-toundary cl' :he :.a:chmenf area ;J ---;-boundary er a part of' the C{jt~~'tJnt Gdnl F/::{'~"- "'-\. brook. lake .L>- ;<.ICiep gauge f Cs6kv6r / '\ == hign;'Jcy, r:ai!l,o/ay "- \ ';;drometeoraiogicai stations Fig. 1. The catchment area of Lake Yelence. Parts of the catchment area: catchment area of the brook Csaszarvfz ...... 400 km~: catchment area of the brook Vereb-Pazmand ..... 123 km": catchment area of the northern shore of Lake Velence 43 km2: catchment area of the southern shore of Lake Velence ... ,1,9 km~: area of Lake Velence at medium water level .•.. 255 km~ The area around Lake Velence is appreciated for its recreational value in Hungary. Recreation aspects require to maintain the lake level constant within narrow limits. Deviation from this optimum level is likely to cause damages, either direct ones (inundation, damages to buildings at high levels, or biological, water quality deterioration at low levels) or indirect ones (in ferior recreation conditions). Since under natural conditions the lake level fluctuated "\videly around the ideal value, artificial regulation has been consid ered. The two reservoirs at Zamoly and Patka have been constructed for this REGL"LATIOS JIODEL 149 purpose. For operating the gates in the reservoirs, and at the outlet from Lake Velence, suitable instructions had to he elaborated. The water released from the series-connected reservoirs is conveyed into the recipient Lake Velence by a hrook called Csaszarvfz. The excess water from the lake is released through the gate at Dinnyes, capahle of discharging up to 4 cu' misec. 60 per cent of the lake surface is covered hy reed. Mean yalue of the precipitation on the lake surface is P = 600 mm, the inflo'w 1= 900 mm, evaporation from the lake E = 950 mm and outflow amounts to 0 = 550 mm. The mean 'water budget is balanced: P..:,-I E-O (1) but shortages are frequent in the catchment. An extensive network of precipita tion gauge,. has been estahlished (with records back to 1900), discharge measure ments started in 1952 at Pakozd on the Csaszarviz brook and in 1963 at Kapolnasnyek on the Vereb-Pazmand brook, the second major watercourse in the catchment. The area drained hy the Csaszarvlz brook down to the Piikozd gauge is 370 sq' km, whereas the catchment area of the Vereb Pazmand brook aboYe the Kapolnasnyek gauge is 113 sq' km. [7]. The ollerational model The model is intended to minimize the losses anticipated to occur as a consequence of undesirahle low and high leyels in Lake Velence. The basic data in the model include the technical data of the reservoirs, such as outlet capacity, highest and lowest levels, etc., further the time series of random changes in the resen-oirs P+R-E where LlW- is the monthly change in water level, P is the monthly precipita tion, R is the monthly runoff, and E is the monthly eyaporation. Values of L1 W- were obtained by direct observations on Lake Velence, or hy simulation using a hydrological runoff model [7], also implying the quadratic loss function based on economic analyses for Lake Velence. This specifies the relative dam ages hy months for periods with low or high stages, in terms of the deviation from the level considered as ideal. With these hasic data the operational model is formulated using the graph in Fig. 2 [9]. Adopting an arhitrary starting month (Month 1), the water yolume stored in the i-th reserYoir at the heginning of the k-th month is Ti(k), 'while the release from the i-th reservoir during the k-th month is ri(k). Similarly LlWi(k) denotes the changes in water level during the k-th month (natural inflow, evaporation-seepage losses), considered as random variahles. 150 KORIS-NAGY DinnYEs-Kajtor canal Csaszar brook 'I t EDIt. r-'----.......,Bella brook! Lapos brook 7 Csasz6r brook I '--,,-----t---,--~-o Lake Szaraz brook I/elence 2ikcvo!gy brook Halaslo canal ) o Rovakja brook [son/rei brook C.::ibulka brook ! lIereb-Pcizrndndi broc.K 1 r 0 -1-[__-=----0- Fig. 2. Diagram of the water system of Lake Velence. Legend: /::. water gauge:. iI sluice According to the equation of continuity --L --L --L T.(kI I 1) -T.(k) 1 J LlW·(k)I I r·1-1 (k) ri(k) (3) (i = 1, 2, 3; k = 1, 2, ... , 12). Here the "water volume released from the (i - l)-th reservoir during the month is assumed to reach the i-th reservoir by the beginning of the next month. From Eq. (3) we obtain: 2 2 2 Ti(3) = Ti(l) + ;Z LlWi(s) + ;Z r,-l - ~ r;(s) 5~1 5~1 s~l and in general: k k k Ti(k) = T;(l) + ;Z LlW,(s) + ;Z r,_l(s) - ~ r,(s) (4) s~l 5~1 5=1 (i = 1, 2, 3, k = 1, 2, . ~ 12). REGULATION JfODEL 151 It is desired to maintain the "water level in Lake Velence around the value considered as ideal. Specifically, the criterion of the optimum is formulated as follows: For given starting water supplies Ti(l) (i = 1, 2, 3), the releases ri(k) minimize the expected value of the actual monthly water volumes in Lake Velence relative to some constant monthly water volume Vii (the ideal water volume). Mathematically (5) Here the "weighting numbers Ih(lh3 = el) express the economic consideration that the same deviation from the optimum water volume is likely to cause different damages in different months. ei : ek = p i. e. damage incurred in the i-th month is p times that in the k-th month. Weighting numbers ek are thus dimensionless ratios and the problem in Eq. (5) is essentially to minimize the quadratic loss function. /{ Introducing the notation Cfi,k = ::E L1Wi(s), Eq. (5) is rewritten: 5=1 (6) For the decision variables, Eq. (6) represents a quadratic programming problem [10] where the quadratic part is and its matrix is, therefore, semi-definite. In solving Eq. (6) it is advisable and expedient to satisfy the following constraints: r. (i = 1, 2, 3; k = 1, 2, ... , 12) (7) where Li denotes the maximum capacity of the outlet regulator of the i-th reservoir for a full month's duration. n. (i = 1, 2, 3; k = 2, 3, ... , 13) (8) where Simin and Simax denote the minimum and maximum water volumes, respectively, to be stored in the i-th reservoir. For the Zamoly and Patka reservoirs, these data are obtained from the physical characteristics, such as the minimum storage volume required for biological life, while in the case of Lake Velence, as the levels desired to be maintained, determined "\vith a view on the actual interests. 152 KGRIS-i\AGY Ill. The additional condition to be met: r;(k) ::::;: T;(k) + ri_l(k) + .JWi(k) - Simin (i = 1. 2, 3; k = 1. 2, . ... 12) written into the form: is thus satisfied, once (3) is. With Eq. (4), (3) becomes I: I: S;ll1~n<T;(I)--;- ;2ri-1(s) - :5'1";(s) (9) s~l s~l (i = 1. 2. 3: k = 1. 2, ... , 12) and since (Pi le is a random variahle, in the case of Lake Velence and the related reserYoir sy~tem the constraints (9) can only he met with certain probahilities (/.;, I: ,3u : i. e., proyidecl: (/.i,l: (lOa) (i = I, 2. 3: k = 1. 2, ... , 12) (lOb) For this reason the problem is that of a chance constrained stochastic qlladratic programming [12, 13, 14, 15]. The proceEs of optimization is as follo"'ws. The quantities rp;" Vie are specified on cconomic considerations, ·while the probability leyels (/.i,/( and /Ji,/( so as to permit compliance with the St:ts (lOa) and (lOb).